概率论基础

1. 条件概率：[Pleft( {B|A} ight) = frac{{P(A,B)}}{{P(A)}}]
2. 乘法定理：[egin{array}{l}
   Pleft( {A,B} ight) = Pleft( {B|A} ight)Pleft( A ight)\
   Pleft( {A,B,C} ight) = Pleft( {C|A,B} ight)Pleft( {B|A} ight)Pleft( A ight)
   end{array}]
3. 全概率公式：[Pleft( A ight) = sumlimits_j {Pleft( {A|{B_j}} ight)Pleft( {{B_j}} ight)} ]
4. 贝叶斯公式：[Pleft( {{B_i}|A} ight) = frac{{Pleft( {A|{B_i}} ight)Pleft( {{B_i}} ight)}}{{sumlimits_j {Pleft( {A|{B_j}} ight)Pleft( {{B_j}} ight)} }}]
5. A，B独立：[Pleft( {A,B} ight) = Pleft( A ight)Pleft( B ight)]
6. 概率分布函数与概率密度函数：[egin{array}{l}
   Fleft( x ight) = Pleft( {X le x} ight) = int_{ - infty }^x {fleft( t ight)} \
   fleft( x ight) = {F^`}left( x ight)
   end{array}]
7. 期望：[Eleft( x ight) = int_{ - infty }^infty  {xfleft( x ight)dx} ]
8. 方差：[Dleft( X ight) = Eleft{ {{{left[ {X - Eleft( X ight)} ight]}^2}} ight} = Eleft[ {{X^2}} ight] - {left[ {Eleft( X ight)} ight]^2}]
9. 协方差：[Covleft( {X,Y} ight) = Eleft{ {left[ {X - Eleft( X ight)} ight]left[ {Y - Eleft( Y ight)} ight]} ight} = Eleft( {XY} ight) - Eleft( X ight)Eleft( Y ight)]
10. 相关系数：[{ ho _{XY}} = frac{{Covleft( {X,Y} ight)}}{{sqrt {Dleft( X ight)} sqrt {Dleft( Y ight)} }}]